Rowan Williams wrote in the New Statesman that 'we are being committed to radical, long-term policies for which no one voted'. Photograph: Steve Parsons/PA

David Cameron has rejected the archbishop of Canterbury's claim that the coalition government is forcing through "radical policies for which no one voted". The prime minister said Rowan Williams was free to express his concerns, but he "profoundly disagreed" with many of his comments.

Conservative and Liberal Democrat cabinet ministers joined backbenchers in registering surprise at the sweep and the specifics of the archbishop's criticisms.

Speaking at a press conference on a visit to Northern Ireland, Cameron said: "I think the archbishop of Canterbury is entirely free to express political views. I have never been one to say that the Church should fight shy of making political interventions.

"But what I would say is that I profoundly disagree with many of the views that he has expressed, particularly on issues like debt and welfare and education."

Williams is guest editor of this week's New Statesman and in an editorial he wrote: "With remarkable speed, we are being committed to radical, long-term policies for which no one voted.

"At the very least, there is an understandable anxiety about what democracy means in such a context."

He criticised the government for continuing to blame the country's difficulties entirely on the deficit it inherited from Labour and said there was "bafflement and indignation" over coalition plans to reform the health service and education.

Vince Cable, the business secretary, said he was equally baffled by Williams's comments.

"The two parties of the coalition got substantially more than half the total vote at the last election and the public knew that we were going to have to embark on very difficult changes, connected with sorting out the massive budget deficit problem," he told Sky News.

He added: "The point which he seemed to be making was that there wasn't enough debate around health reform, for example, which I don't understand because there's a very big debate. My party has triggered it, we're having a pause, rethinking the reforms. So he's obviously had his views and it's welcome that he pitches into political debate but I think he's actually wrong on the specifics."

The welfare secretary, Iain Duncan Smith, defended himself against an accusation by the archbishop that he brought back "the seductive language of 'deserving' and 'undeserving' poor". Williams criticised "the steady pressure" to increase "punitive responses to alleged abuses of the system".

Duncan Smith said the archbishop should have been more balanced in his comments.

The public would have been more anxious about the coalition if they had not tackled benefit dependency, Duncan Smith told the BBC.

"With respect to the archbishop of Canterbury, I have never ever spoken about the deserving or undeserving poor. I don't believe in that concept. All I say is that the system has created an undeserving group, that's what it has created," he said.

Ministers were also surprised at the archbishop's suggestion that education reforms had not been well-trailed before and during the election. Nick Boles, the private secretary of the education minister Nick Gibb, suggested that church schools were acting as a producer interest, trying to prevent the spread of academy schools.

In the article, Williams accepted that the government's big society agenda was not a "cynical walking-away from the problem". But he warned there was confusion about how voluntary organisations would "pick up the responsibilities shed by government", and said that the big society was seen with "widespread suspicion".

"The uncomfortable truth is that, while grassroots initiatives and local mutualism are to be found flourishing in a great many places, they have been weakened by several decades of cultural fragmentation," Williams wrote.

He also criticised the chancellor, George Osborne, saying: "It isn't enough to respond with what sounds like a mixture of, 'this is the last government's legacy' and 'we'd like to do more, but just wait until the economy recovers a bit'."

Williams also singled out Labour for failing to produce fresh ideas since going into opposition. Westminster politics "feels pretty stuck", he warned, adding that his aim was to stimulate a livelier debate and to challenge the left to develop its own "big idea" as an alternative to the Conservative-Liberal Democrat alliance.

Andy Burnham, Labour's shadow education secretary, ducked this element of the piece and instead said people would share Williams's concerns about the government's pursuit of policies for which it has no mandate.

Burnham said: "This government has no mandate for cutting too far and too fast, subjecting the NHS to a reckless top-down reorganisation and launching an unprecedented attack on young people by scrapping EMA and trebling tuition fees."

Lord Tebbit, former Conservative chairman and cabinet minister, said it was part of the archbishop of Canterbury's job to "make comments of a political kind in this area". Tebbit, a critic of the coalition, told Today that Williams was highlighting a "problem of coalition".

But Gary Streeter, chair of the all-party Christians in Parliament group, said: "I think the people are with us on this and the archbishop, sadly and unusually for him, has ill-judged his attack."

Downing Street tried to avoid a Church v state row by stressing the Archbishop was entitled to speak out, even if ministers were specially surprised at his suggestion that education secretary Michael Gove's education reforms had not been well trailed well before and during the election.

Nick Boles, the private secretary of the education minister Nick Gibb, suggested that church schools were acting as a producer interest, trying to prevent the spread of academy schools.